Lee-Yang theory, high cumulants, and large-deviation statistics of the magnetization in the Ising model

TL;DR

This paper introduces a cumulant method to determine Lee-Yang zeros from magnetization fluctuations in the Ising model, enabling phase transition predictions and insights into rare magnetization fluctuations.

Contribution

The study develops a cumulant-based approach to extract Lee-Yang zeros from small system fluctuations, linking them to large-deviation functions and phase transition analysis.

Findings

Successfully determined Lee-Yang zeros in 1D, 2D, and 3D Ising models.

Predicted phase transition points from finite-size data.

Connected Lee-Yang zeros to the large-deviation function of magnetization.

Abstract

We investigate the Ising model in one, two, and three dimensions using a cumulant method that allows us to determine the Lee-Yang zeros from the magnetization fluctuations in small lattices. By doing so with increasing system size, we are able to determine the convergence point of the Lee-Yang zeros in the thermodynamic limit and thereby predict the occurrence of a phase transition. The cumulant method is attractive from an experimental point of view since it uses fluctuations of measurable quantities, such as the magnetization in a spin lattice, and it can be applied to a variety of equilibrium and non-equilibrium problems. We show that the Lee-Yang zeros encode important information about the rare fluctuations of the magnetization. Specifically, by using a simple ansatz for the free energy, we express the large-deviation function of the magnetization in terms of Lee-Yang zeros. This result may hold for many systems that exhibit a first-order phase transition.